非线性最优控制问题的保辛多层次求解方法

Symplectic Multi-Level Method for Solving Nonlinear Optimal Control Problem

将非线性系统的最优控制问题导向Hamilton系统,提出了求解非线性最优控制问题的保辛多层次方法.首先,以时间区段两端状态为独立变量并在区段内采用Lagrange插值近似状态和协态变量,通过对偶变量变分原理将非线性最优控制问题转化为非线性方程组的求解.然后,在保辛算法的具体实施过程中提出了多层次求解思想,以2N类算法为基础由低层次到高层次加密离散时间区段,利用Lagrange插值得到网格加密后的初始状态与协态变量作为求解非线性方程组的初值,可提高计算效率.数值算例验证了算法在求解效率与求解精度上的有效性.

The optimal control problem for nonlinear system was transformed into Hamiltonian system and a symplectic-preserving method was proposed. The state and costate variables were approximated by Lagrange polynomial and state variables at two ends of the time interval were taken as the independent variables, and then based on the dual variable principle, nonlinear op- timal control problems were replaced by nonlinear equations. In the implement of symplectic al- gorithm, based on the 2N algorithm, a mnlti-level method was proposed. When the time grid was refined from the low level to the high level, the initial state variables and costate variables of nonlinear equations could be obtained from Lagrange interpolation at the low level grid, which could improve the efficiency. Numerical simulations show the precision and efficiency of the proposed algorithm.